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MIT6_851S10_assn08_sol - v i and vertices in B Otherwise we...

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6.851 Advanced Data Structures (Spring’10) Prof. Erik Demaine Dr. Andr´ e Schulz TA: Aleksandar Zlateski Problem 8 Sample Solutions Cuckoo Hashing. Consider a cuckoo graph with t edges. We have the probability of any partic- ular cuckoo graph to be 2 2 t t . There are n t m t configurations of that graph that contain a cycle m (there must be a selection of n elements with their hash values on the cycle). Hence, the probability that a cuckoo graph with t edges contains a cycle is n t m m 2 t t 2 t = 3 1 t . Since 1 = 1 , we conclude that the probability of a cuckoo graph containing no cycles is at least t =2 3 t 2 1 / 2. Conditional Expectations. Consider the following greedy algorithm. We start with two empty sets A and B . Now we consider all the vertices one by one. For each vertex v i we decide to add it to the set A if there are less edges between v i and vertices in A then edges between
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Unformatted text preview: v i and vertices in B . Otherwise, we add v i to B . This is equivalent to choosing where to put v i by maximizing the conditional expectation on the size of the cut given A ⊆ V and B ⊆ V \ V . This way we will end up with at least half of the edges having one vertex in A the other in B . Setting V = A we get a cut with the value ≥ | E / 2. | 1 MIT OpenCourseWare http://ocw.mit.edu 6.851 Advanced Data Structures Spring 2010 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms ....
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